The ultimate goal of this chapter is to present Malliavin’s theorem on the failure of spectral synthesis in the L 1 algebra of a noncompact locally compact Abelian group. As usual, we find many an interesting byway as we pursue the constructions needed for Malliavin’s theorem. In § 39, we look at commutative Banach algebras and their closed ideals, slipping in some classical analysis en route. Section 40 deals with spectra for bounded functions and with certain spectral sets. It is rather technical but possesses at least the merit of brevity. In § 41, we construct some pathological sets required in our treatment of Malliavin’s theorem, and also look at some other sets possessing curious analytic properties. Section 42, the last of the chapter, completes the proof of Malliavin’s theorem.
KeywordsClosed Subset Invariant Subspace Banach Algebra Structure Space Compact Abelian Group
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